Problem: The sum of two angles is $89^\circ$. Angle 2 is $83^\circ$ smaller than $3$ times angle 1. What are the measures of the two angles in degrees?
Let $x$ equal the measure of angle 1 and $y$ equal the measure of angle 2. The system of equations is then: ${x+y = 89}$ ${y = 3x-83}$ Since we already have solved for $y$ in terms of $x$ , we can use substitution to solve for $x$ and $y$ Substitute ${3x-83}$ for $y$ in the first equation. ${x + }{(3x-83)}{= 89}$ Simplify and solve for $x$ $ x+3x - 83 = 89 $ $ 4x-83 = 89 $ $ 4x = 172 $ $ x = \dfrac{172}{4} $ ${x = 43}$ Now that you know ${x = 43}$ , plug it back into $ {y = 3x-83}$ to find $y$ ${y = 3}{(43)}{ - 83}$ $y = 129 - 83$ ${y = 46}$ You can also plug ${x = 43}$ into $ {x+y = 89}$ and get the same answer for $y$ ${(43)}{ + y = 89}$ ${y = 46}$ The measure of angle 1 is $43^\circ$ and the measure of angle 2 is $46^\circ$.